Figure 9.10 depicts the generic operation of filtering
by
to produce
, where
is the impulse response of the
filter. The output signal is given by the convolution of
and
:

In this form, it is clear why the filter (9.5) is called
``running sum'' filter. Dividing it by
, it becomes a ``moving
average'' filter, averaging the most recent
input samples. It is
also called an integrated comb filter.10.1

Figure 9.11 shows the amplitude response of the running-sum
lowpass filter for length
. The gain at dc is
, and nulls
occur at
and
. These nulls occur
at the sinusoidal frequencies having respectively one and two periods
under the 5-sample ``rectangular window''. (Three periods would need
at least
samples, so
doesn't ``fit''.) Since
the pass-band about dc is not flat, it is better to call this a
``dc-pass filter'' rather than a ``lowpass filter.'' We could also
call it a dc sampling filter.10.2